47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition
5 - 8 January 2009, Orlando, Florida
AIAA 2009-240
DNS of auto–ignition in turbulent diffusion H2/air
flames
Jeff Doom∗ and Krishnan Mahesh†
University of Minnesota, Minneapolis, MN, 55455, USA
Direct numerical simulation (DNS) is used to study auto–ignition of turbulent
diffusion flames. A novel, all–Mach number algorithm developed by Doom et al1 is
used. The chemical mechanism is a nine species, nineteen reaction mechanism for
H2 and Air from Mueller at el.2 Simulations of three dimensional turbulent diffusion flames are performed. Isotropic turbulence is superimposed on an unstrained
diffusion flame where diluted H2 at ambient temperature interacts with hot air.
Both, unity and non–unity Lewis number are studied. The results are contrasted
to the homogeneous mixture problem and laminar diffusion flames. Results show
that auto–ignition occurs in fuel lean, low vorticity, high temperature regions with
low scalar dissipation around a most reactive mixture fraction, ζM R (Mastorakos
et al.3 ). However, unlike the laminar flame where auto-ignition occurs at ζM R , the
turbulent flame auto–ignites over a very broad range of ζ around ζM R , which cannot
completely predict the onset of ignition. The simulations also study the effects of
three-dimensionality. Past two–dimensional simulations (Mastorakos et al.3 ) show
that when flame fronts collide, extinction occurs. However, our three dimensional
results show that when flame fronts collide; they can either increase in intensity,
combine without any appreciable change in intensity or extinguish. This behavior
is due to the three–dimensionality of the flow.
I.
Introduction
The auto–ignition of turbulent diffusion flames is central to applications such as diesel engines and
scramjet engines where fuel is injected into hot oxidizer. Fuel and oxidizer mix through convection
and diffusion, then auto–ignite due to the high temperatures of the oxidizer. Direct numerical
simulation (DNS) is used to study a diffusion flame where fuel (hydrogen/nitrogen) reacts with air
(oxygen/nitrogen). For laminar flames, fuel and oxidizer diffuse, then react, yielding products. In
turbulent flames, one might see auto–ignition, extinction and even re–ignition due to the turbulence.
Mahalingam et al.4 performed DNS of three-dimensional turbulent non–premixed flames with
finite rate chemistry. They incorporated one–step and two–step reaction models for the chemistry.
Their results show that the intermediate species concentration overshoots the experiments. They
suggest that this behavior is due to time–dependent strain rates in the turbulent flow. Montgomery
et al.5 also performed DNS of a three-dimensional turbulent hydrogen–oxygen non–premixed flame
using a reduced mechanism (7–species, ten–reactions). The simulations were used to suggest a
mixture fraction and progress variable model for the chemical reactions. Im et al.6 performed
DNS of two–dimensional turbulent non–premixed hydrogen–air flames to study auto–ignition. They
show that peak values of HO2 align with maximum scalar dissipation during ignition. Im et al.
also noted that weak and moderate turbulence enhanced auto–ignition while stronger turbulence
delayed ignition. Echekki & Chen7 used DNS to study auto–ignition of a hydrogen/air mixture in
∗ Graduate
† Associate
Research Assistant
Professor
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Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
two dimensional turbulence. Their results show spatially localized sites where auto–ignition begins,
which they define as a “kernel”. These kernels have a build up of radicals at high temperatures, fuel–
lean mixtures, and low dissipation rates. Mastorakos et al. performed two–dimensional simulation
of auto–ignition of laminar and turbulent diffusion flames with one–step chemistry. They found that
ignition always occurs at a well–defined mixture fraction (ζM R ). Hilbert & Thevenin8 performed
similar simulations to Mastorakos et al.3 and Im et al.6 The difference between Hilbert & Thevenin9
and others is that the simulation uses multicomponent diffusion velocities.
A significant difference between the past work cited above, and our DNS is that we perform
fully three dimensional simulations using finite rate chemistry. Also, we incorporate the Mueller
mechanism for hydrogen/air (nine species and nineteen reaction) and not a reduced mechanism.
The objectives of our simulations are: (i) to study major differences between two dimensional and
three dimensional simulations of turbulent diffusion flames. In particular, what happens to the flame
front in three dimensions? (ii) How do the ignition kernels evolve in time? (iii) What is the effect
of unity Lewis number and non–unity Lewis number on ignition, and (iv) How does the mixture
fraction compare to a passive scalar?
This paper is organized as follows. The governing equations and numerical method are described
in section II and section II.A. Section III provides details of the laminar diffusion flame, turbulent
diffusion flame and validation. The simulations results are discussed in section IV. In the results
section, the homogeneous problem (IV.A), auto–ignition (IV.B), Lewis number (IV.C) and flame
front evolution (IV.D) are discussed in detail. A brief summary in section V concludes the paper.
II.
α
(a)
fuel
Governing equations
oxidizer
x
(b)
Figure 1. Schematic of initial conditions for the turbulent diffusion flame. (a) Diffusion flame and (b) Isotropic
turbulence.
The governing equations are the unsteady, compressible, reacting Navier–Stokes equations:
∂ρudj
∂ρd
+
= 0,
d
∂t
∂xdj
(1)
∂ρYkd udj
∂ρd Ykd
∂
+
=
∂td
∂xdj
∂xdj
∂Y d
ρd Dkd kd
∂xj
!
+ ω̇kd ,
d
∂ρd udi udj
∂τij
∂ρd udi
∂pd
+
=
−
+
∂td
∂xdj
∂xdi
∂xdj
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(2)
(3)
∂ ρd E d + pd udj
∂ρd E d
+
∂td
∂xdj
=
d d
∂τij
ui
∂xdj
∂
+
∂xdj
pd = ρd Rd T d = ρd
cdp ∂T d
µ
P r ∂xdj
d
!
+
N
X
Qdk ω̇kd ,
(4)
k=1
Ru d
T .
Wd
(5)
The superscript ‘d’ denotes the dimensional value. From Doom et al.,1 non–dimensional variables
are defined as:
ρd
ud
td
µd
pd − pr
,
, ui = i , t =
, µ=
, p=
ρr
ur
L/ur
µr
ρr ur 2
Td
ur
ur
T =
, pr = ρr Rr Tr ,
, Mr =
=√
Tr
ar
γRr Tr
Yd
Dd
Lω̇kd
Yr Qdk
Yk = k , Dk = k , ω̇k =
, Qk =
,
Yr
ur L
ur ρr Yr
cp,r Tr
ρ=
Rr =
Ru
Sck
Wd
, and W =
, Lek =
.
Wr
Pr
Wr
(6)
(7)
(8)
(9)
Let the subscript ‘r’ denote the reference variable. Note that pressure is non–dimensionalized using
an incompressible scaling motivated by Thompson.10 Therefore, the non–dimensional governing
equations are:1
∂ρ ∂ρuj
+
= 0,
∂t
∂xj
∂ρYk
∂ρYk uj
1
∂
∂Yk
+
=
µ
+ ω̇k ,
∂t
∂xj
ReSck ∂xj
∂xj
∂ρui uj
∂ρui
∂p
1 ∂τij
+
=−
+
,
∂t
∂xj
∂xi
Re ∂xj
γ−1
∂
γ−1
∂
∂uj
ρui ui +
ρui ui uj +
Mr2
p+
γp +
∂t
2
∂xj
2
∂xj
N
X
(γ − 1)Mr2 ∂τij ui
∂
1
µ ∂T
=
Qk ω̇k ,
+
+
Re
∂xj
ReP r ∂xj W ∂xj
k=1
| {z }
(10)
(11)
(12)
(13)
ω˙n
ρT
= γMr2 p + 1.
W
(14)
Here, ρ, T , p, ui and Yk denote non–dimensional density, temperature,pressure, velocities and
mass
∂uj
∂ui
2 ∂uk
fraction of ‘k’ species respectively. The viscous stress tensor τij = µ ∂x
+
−
δ
∂xi
3 ∂xk ij . The
j
source term is denoted as: ω̇k which is modeled
P using the Arrhenius law. The heat of reaction per
unit mass in the energy equation is Qk and
Qk ω̇k or ω̇n is the heat release due chemical reactions.
Sck is the Schmidt number for the k th species, P r is the Prandtl number, and Re is the Reynolds
number. W is the mean molecular weight of the mixture.
II.A.
Numerical method
The numerical algorithm is discussed in detail by Doom et al.1 The algorithm is fully implicit,
spatially non–dissipative and second order in time & space. The thermodynamic variables and
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species are staggered in time from velocity. All variables are co–located in space. A pressure–
correction method is used to enforce velocity divergence obtained from the energy equation. The
discrete equations discretely conserve kinetic energy in the incompressible, non–reacting, inviscid
limit. These features make the algorithm stable and accurate at high Reynolds numbers, and
efficient at both low and finite Mach numbers. These features are attractive for DNS of compressible
turbulent reacting flows.
III.
YH2
Problem statement
T
(a)
time
ω̇n
(b)
x
Figure 2. (a) Comparison of a major species & temperature between Chemkin and present algorithm for
a well–stirred reactor. ([YH2 ]
Algorithm, 4 Chemkin & [T ]
Algorithm,
Chemkin). (b) Grid
convergence study showing heat release (ω̇n ).
III.A.
Laminar diffusion flame
Figure 1 (a) shows a laminar diffusion flame. The reference variables for the diffusion flame are
shown in table 1. Note that Mr , τr , Tr , and ρr are reference Mach number, reference time, reference
temperature, and reference density, respectively. For all simulations, Re is 1000, P r is 0.7 and µ is
equal to T 0.7 . For the unity Lewis number cases, Lek = 1 and the non–unity Lewis numbers are (6 ):
where Lek =
Sck
Pr .
LeH2 = 0.3, LeO2 = 1.11, LeO = 0.7, LeOH = 0.73,
(15)
LeH2 O = 0.83, LeH = 0.18, LeHO2 = 1.10, LeH2 O2 = 1.12
(16)
The initial conditions for the diffusion flame are from Mastorakos et al.:3
YH2 = YH0 2 ζ (x) ,
YO2 = YO02 [1 − ζ (x)] ,
T = TO + [TF − TO ] ζ (x) ,
(x − xc )
α (x) = 0.5 1 − erf
.
δ
(17)
α is the initial conditions for mixture fraction (figure 1 a). Note that δ equal 0.05 which is the
thickness of the mixing layer and xc is the center of the mixing layer where it is equal to π. YH0 2 ,
YO02 , TO , and TF are equal to 0.029, 0.233, 4, and 1, respectively. This will yield an equivalence ratio
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Table 1. Reference variables.
Mr
0.1
τr
1.48e-4 (sec)
Tr
300 (K)
ρr
0.8282 (kg/m3 )
(φ) of one. The non–dimensional pressure was set to zero. Density was obtained using equation
14. The inlet boundary condition are set to a constant and zero derivative boundary conditions are
specified at the outflow.
III.B.
Turbulent diffusion flame
τchem
(a)
τchem
ζ
(b)
ζ
Figure 3. (a) Mixture fraction versus τchem for varying temperature in kelvin.
φ = 1, T = 1100,
φ = 1, T = 1200,
φ = 1, T = 1300, and
φ = 1, T = 1400. (b) Mixture fraction versus τchem for
φ = 1/4, T = 1200,
φ = 1, T = 1200,
φ = 4, T = 1200 and
φ = 8,
varying fuel/air ratio,
T = 1200.
Three dimensional isotropic turbulence (figure 1 b) is superimposed on the laminar diffusion
flame. The initial isotropic turbulence field is prescribed by the three–dimensional kinetic energy
spectrum:
" #
r
4
2
2 u20 k
k
E (k) = 16
exp −2
.
(18)
π k0 k0
k0
The turbulent Reynolds number Reλ is 50 and k0 is 5. The computational domain is 2π for x, y, and
z. The computational grids for the turbulent diffusion flame are 128 and 256 cubed. All turbulent
diffusion flame figures are 128 cubed excect figure 9 which is 256 cubed.
III.C.
Validation
The homogeneous mixture problem is used to validate the complex chemistry. Figure 2 (a) is a
comparison to Chemkin.11 Good agreement is observed for the homogeneous mixture problem. A
grid convergence study was performed for the laminar diffusion flame. Figure 2 (b) shows results
from the grid convergence study. Uniform grids ranging from 64 to 2048 points were used. A
grid–converged solution is obtained for a grid of 128 points.
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IV.
IV.A.
Results
Homogeneous mixture problem
Mastorakos et al.3 used one–step chemistry and the homogeneous mixture problem to obtain τchem
and the mixture fraction most likely to auto–ignite (ζM R ). We obtain ζM R for the Mueller mechanism
(Mueller et al.2 ). The chemical mechanism is a 9 species (H2 , O2 , OH, O, H, H2 O, HO2 , H2 O2 ,
and N2 ), 19 reaction mechanism. Figure 3 (a) shows mixture fraction versus τchem for initial
temperatures that range from 1100 to 1400 Kelvin and figure 3 (b) shows mixture fraction versus
τchem for fuel/air ratios (φ) that range from 0.25 to 8. Note in figure 3 that the minimum τchem
corresponds to ζM R . Figure 3 also shows the effect of fuel/air mixture and temperature on τchem .
As temperature increases, τchem becomes smaller and ζM R shifts to the right. As φ increases, τchem
becomes smaller and ζM R shifts to the left. These results illustrate the balance between adequate
fuel and high enough temperature for auto–ignition to occur. Table 2 lists the most reactive mixture
fraction and corresponding values of the initial temperature, fuel/air ratio and τchem . Oxidizer
temperature of 1200 K (T0 ) and φ of one is used in the turbulent simulations.
Table 2. Results of homogeneous problem.
T0
1100
1200
1300
1400
1200
1200
1200
IV.B.
φ
1
1
1
1
1/4
4
8
ζM R
0.06
0.08
0.09
0.10
0.1
0.06
0.04
τchem
3.4e − 4
1.48e − 4
8.5e − 5
5.6e − 5
3.5e − 4
8.0e − 5
6.4e − 5
TM R
1052
1128
1210
1290
1110
1146
1164
Auto–ignition of diffusion flame
This section discusses auto–ignition of the diffusion flame and its relation to temperature and mixture
fraction. The passive scalar and mixture fractions used in the simulations are defined as:
∂ρZ
∂ρZuj
1
∂
∂Z
+
=
µ
,
(19)
∂t
∂xj
ReScZ ∂xj
∂xj
YF
YO
1
φ 0 − 0 +1 ,
(20)
ζP =
φ+1
YF
YO
YH
2YH2 O
YOH
YHO2
2YH2 O2
2YH2
Z H = WH
+
+
+
+
+
,
(21)
WH2
WH
WH2 O
WOH
WHO2
WH2 O2
2YO2
YO
YH2 O
YOH
2YHO2
2YH2 O2
Z O = WO
+
+
+
+
+
,
WO 2
WO
WH2 O
WOH
WHO2
WH2 O 2
1
ZH /WH + YO02 − ZO /WO
ζB = 2 1 0
.
0
2 YH2 /WH + YO2 /WH
Here Z denotes the passive scalar variable. ζP and ζB denote mixture fractions from Poinsot &
Veynante12 and Hilbert & Thevenin,8 respectively. Note that ζB includes all species where ζP
accounts for only fuel and oxidizer. For the passive scalar equation, ScZ was chosen to be 0.7.
Figure 4 is scatter plot of the laminar diffusion flame for unity and non–unity Lewis number. If the
passive scalar and mixture fraction were equal, one would obtain a straight line. Note that the unity
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ζB
ζP
(a)
ζB
ζP
Z
(b)
Z
Figure 4. (a) scatter plot of the laminar diffusion flame with unity Lewis number mixture fraction ( ζB and
ζP ) versus passive scalar (Z) from a non–dimensional time of zero to 10. (b) scatter plot of the laminar
ζP ) versus passive scalar (Z) from
diffusion flame with non–unity Lewis number mixture fraction ( ζB and
a non–dimensional time of zero to 10.
Lewis number yields a straight line but the non–unity Lewis number does not. These results show
that mixture fraction (as defined above) is a conserved scalar for the unity Lewis number but not
the non–unity Lewis number. Figure 4 also shows the slight differences between ζP and ζB , observed
well after the formation of products. Therefore, ζP is adequate for studying auto–ignition and from
now on, ζ = ζP .
Figure 5 (a) illustrates the evolution of ignition for the laminar diffusion flame. Note that
ignition occurs in fuel lean, high temperature regions (ζ = ζM R = 0.08) and propagates towards the
stoichiometric side (ζ = 0.5). Unity Lewis number behaves in a similar manner, but ignition occurs
at a later time (figure 5 a). Recall that for unity Lewis number, the mixture fraction is a passive
scalar. Therefore, auto–ignition occurs in the same region (ζM R ). On the other hand, for non–unity
Lewis number, auto–ignition occur at ζP = ζB = 0.08 for the mixture fractions and Z = 0.13 for the
passive scalar. This result shows that auto–ignition occurs in a ‘richer’ region for the passive scalar
equation than the mixture fraction for non–unity Lewis number.
For the turbulent diffusion flame, figures 6, 7 and 8 show the evolution of T , YHO2 , and YOH ,
respectively. Each contour plot is separated by a unit of time from zero to four. Figure 6 illustrates
the effect of auto–ignition on temperature. In figures 7 and 8, note the differences between YHO2
and YOH , especially the range of length scales.
Contour plots (in the x = π plane) of ω̇n , T , ζ2 , χ2 , YHO2 , and |ω| at a non–dimensional time of
0.9 for the turbulent diffusion flame are shown in figure 9. The contour lines represent the mixture
fraction over the range from 0.05 to 0.30. Note that ignition occurs in fuel lean regions (9 a).
Echekki and Chen define such ignition sites as kernels. They describe the kernels sites as regions of
high–temperature (9 b), fuel–lean mixtures (9 (c)) and low dissipation rates (9 (d)) which is present
in figure 9. Note that scalar dissipation is defined as:
∂ζ ∂ζ
2
.
(22)
χ=
Re ∂xj ∂xj
In the turbulent diffusion flame, there will also be regions with low vorticity (figure 9 e). The
ignition sites have low vorticity which allows time for fuel and oxidizer to diffuse together to form
essential products for auto–ignition to occur e.g. HO2 . The formation of HO2 is the precursor to
auto–ignition (Im et al.6 ) and is shown in figure 9 (f).
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Ignition (ζM R )
ω̇n
(a)
T
ω̇n
ζ
(b)
time
Figure 5. (a) evolution of mixture fraction versus heat release and the location of ignition ζM R . (b) temporal
temperature
evolution of maximum heat release and temperature for the laminar diffusion flame.
of unity Lewis number,
temperature of non–unity Lewis number,
heat release of unity Lewis
number, and
heat release of non–unity Lewis number.
Figure 6. Evolution of temperature from a non–dimensional time of 0 to 4.
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Figure 7. Evolution of YHO2 from a non–dimensional time of 0 to 4.
Figure 8. Evolution of YOH from a non–dimensional time of 0 to 4.
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ω̇n
T
(a)
(b)
ζP
χ
(c)
(d)
|ω|
(e)
YHO2
(f)
Figure 9. Contour plot of ω̇n , T , ζ, χ2 , YHO2 , and |ω| at a non–dimensional time of 0.9. Contour lines are
superimposed for the mixture fraction ζ and range from 0.05 to 0.30.
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T
(a)
T
ζ
(b)
ζ
Figure 10. Plot of mixture fraction (ζ) versus Temperature in Kelvin (T ) (a) laminar diffusion flame and (b)
is the turbulent diffusion flame.
Figure 10 shows the effect of mixing on auto–ignition at ζM R . Recall from the laminar diffusion
flame that auto–ignition occurred at ζ = ζM R = 0.08 (figure 5 a). The homogeneous mixture
problem predicted the onset of ignition (ζM R ) very well for the laminar diffusion flames. On the
other hand, ζM R does not entirely predict the onset of ignition for the turbulent diffusion flame.
This phenomenon is completely due to the turbulent mixing which results in a range of temperatures
for a given mixture fraction. For example, if ζ = 0.2, then temperature ranges from 950 K to 1050 K
(figure 10 b). Recall from the homogeneous mixture problem, that temperature plays an important
role in auto–ignition. Table 2 illustrate that a 100 Kelvin difference changes τchem by a factor of
2.3. For the turbulence case, this can greatly affect the onset of ignition and location. This is why
auto–ignition occurs in a very broad range for ζM R in the turbulent diffusion flame while the laminar
case is at ζM R .
These results suggest that maybe one should track temperature instead of mixture fraction
in predicting auto–ignition and the evolution of ignition. Figure 11 superposes contour lines of
temperature which are chosen to range from 950 to 1150 K (TM R = 1200 − ζM R [1200 − 300]).
Interestingly, maximum heat release follows TM R at every instant of time in figure 11. This figure
shows that the onset of ignition occurs at ζM R and TM R . As ignition continues, ζM R is in the very
fuel lean region and does not follow the path of ignition. TM R on the other hand, follows the path
of ignition at each instance in time. This suggests that TM R instead of ζM R might be better choice
in predicting the evolution of auto–ignition.
IV.C.
The effect of Lewis number
Figure 5 (a) shows the profiles of maximum heat release and temperature versus time for the laminar
diffusion flame. Note that ignition occurs at a non–dimensional time of around 1.6 for the one
dimensional diffusion flame with unity Lewis number while ignition occurs around 1 for the non–
unity Lewis number. The temperature difference between the unity Lewis number and non–unity
Lewis number is 450 K. The fastest ignition time was obtained for the non–unity Lewis number
turbulent diffusion flame. The first signs of auto–ignition starts around 1.6, 1.0, 1.4 and 0.9 (non–
dimensional time) for the laminar unity Lewis number, laminar non–unity Lewis number, turbulent
unity Lewis number and turbulent non–unity Lewis number, respectively. These results shows that
differential diffusion plays an important in the auto–ignition of the diffusion flame. The difference
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ζ at time = 1.8
T at time = 1.8
ζ at time = 2.2
T at time = 2.2
ζ at time = 2.6
T at time = 2.6
Figure 11. Contour lines of mixture fraction (0.05 to 0.3) and temperature (950 to 1150 K) superposed on
shaded contours of reaction rate at different instants of time.
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(a)
(b)
Figure 12. Comparison of temperature for (a) unity Lewis number and (b) non–unity Lewis number at a
non–dimensional time of 3.
(a)
(b)
(c)
Figure 13. Contour plots of ω̇n showing time evolution of flame fronts. (a) time = 2.6, (b) time = 2.9, and
(c) time = 3.2.
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between the turbulent unity Lewis number and non–unity Lewis number is further demonstrated in
figure 12. The comparison between temperature in figure 12 shows that the two cases are complete
different, i.e. viscous diffusion is important even in the turbulent regime.
IV.D.
Evolution of ignition fronts
It is generally felt (e.g. Hilbert & Thevenin8 and Mahalingam et al.4 ) that the main difference
between three dimensional and two dimensional turbulent simulation is that two dimensional turbulence lacks vortex stretching and high dissipation rates. Our results suggest that three-dimensional
simulations do matter in the collision of flame front and that dimensionality affects the results at a
more fundamental level. Mastorakos et al3 performed two-dimensional simulations using one–step
chemistry. They found that fronts propagate outward and collapse at the very lean or rich sides due
to extinction, or after collisions with separate similar fronts originating from other ignition sites.3
Our results show that when flame fronts collide; they can either increase in intensity, combine without any appreciable change in intensity, or extinguish. In most cases, the flame fronts combine
together (figure 13). This phenomenon is due to the extra dimension. In two–dimensional simulations when flame fronts collide, there is no extra dimension for products to travel which ‘chokes’
the flow. In the three-dimensional simulation, products can migrate to the extra dimension which
allows the flame fronts to combine together.
V.
Summary
The paper uses direct numerical simulation to study auto–ignition of a hydrogen/air turbulent
diffusion flame. A detailed chemical mechanism due to Mueller et al2 is used. Simulations of three
dimensional turbulent diffusion flames were performed. Isotropic turbulence is superimposed on an
unstrained diffusion flame where diluted H2 at ambient temperature interacts with hot air. Both,
unity and non–unity Lewis number are studied. The results are contrasted to the homogeneous
mixture problem and a laminar diffusion flames. Results show that auto–ignition occurs in fuel lean,
low vorticity, and high temperature regions with low scalar dissipation around ‘most reactive’ mixture
fraction, ζM R . For the laminar case, auto-ignition occurs at ζM R while the turbulent case auto–
ignites occurs in a very broad range of ζM R . Simulation also study the effects of three-dimensionality
of the turbulent diffusion flame. Two–dimensional simulations show that when flame fronts collide,
extinction occurs (Mastorakos et al.3 ). Our results show that when flame front collide; they can
either increase in intensity, combine together or go extinct which is due to the extra dimension. Three
dimensional simulations are essential to turbulent diffusion flames and in particular, the collision of
flame fronts. Ignition kernels are related to the most reactive mixture fraction, but the most reactive
fraction cannot completely predict the onset of ignition because ignition occurs in a broad range of
ζM R . We also show that the passive scalar equation does not agree with non–unity Lewis number
mixture fraction and that viscous diffusion is important even in the turbulent regime.
VI.
Acknowledgments
This work was supported by the AFOSR under grant FA9550–04–1–0341 and by the Doctoral Dissertation fellowship awarded by the Graduate School at the University of Minnesota. Computing resources were provided by the Minnesota Supercomputing Institute, the San Diego Super–computing
Center and the National Center for Supercomputing Applications.
References
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226 (2007) 1136–1151.
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2 M.A. Mueller, T.J. Kim, R.A. Yetter, F.L. Dryer, Flow Reactor Studies and Kinetic Modeling of the H2/O2
Reaction. Int. J. Chem. Kinet. 31 (1999) 113–125.
3 E. Mastorakos, T. A. Baritaud, T. J. Poinsot, Numerical simulations of autoignition in turbulent mixing flows.
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